Linear Algebra Done Right · Problem 7F.10

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · operator-norms-invertibility

Problem

Suppose

\dim V > 1

and

T \in \mathcal{L}(V)

is not invertible. Prove that for every

\epsilon > 0

, there exists

S \in \mathcal{L}(V)

such that

0 < \|T - S\| < \epsilon

and

S

is not invertible.